Introduction Recent Developments in Classical Density Functional Theory: Internal Energy Functional and Diagrammatic Sturucture of Fundamenta

arXiv:1212.6358v1 [cond-mat.soft] 27 Dec 2012 Condensed Matter Physics, 2012, Vol. 15, No 4, 43603: 1–15 DOI: 10.5488CMP.15.43603 http:www.icmp.lviv.uajournal Recent developments in classical density functional theory: Internal energy functional and diagrammatic structure of fundamental measure theory M. Schmidt 1,2 , M. Burgis 1 , W.S.B. Dwandaru 2,3 , G. Leithall 2 , P. Hopkins 2 1 Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D–95440 Bayreuth, Germany 2 H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK 3 Jurusan Fisika, Universitas Negeri Yogyakarta, Bulaksumur, Yogyakarta, Indonesia Received August 15, 2012 An overview of several recent developments in density functional theory for classical inhomogeneous liquids is given. We show how Levy’s constrained search method can be used to derive the variational principle that un- derlies density functional theory. An advantage of the method is that the Helmholtz free energy as a functional of a trial one-body density is given as an explicit expression, without reference to an external potential as is the case in the standard Mermin-Evans proof by reductio ad absurdum. We show how to generalize the approach in order to express the internal energy as a functional of the one-body density distribution and of the local entropy distribution. Here the local chemical potential and the bulk temperature play the role of Lagrange mul- tipliers in the Euler-Lagrange equations for minimiziation of the functional. As an explicit approximation for the free-energy functional for hard sphere mixtures, the diagrammatic structure of Rosenfeld’s fundamental mea- sure density functional is laid out. Recent extensions, based on the Kierlik-Rosinberg scalar weight functions, to binary and ternary non-additive hard sphere mixtures are described. PACS: 61.25.-f, 61.20.Gy, 64.70.Ja Key words: density functional theory, Hohenberg-Kohn theorem, Rosenfeld functional

1. Introduction

The theoretical study of inhomogeneous classical liquids received a boost through the development of classical density functional theory DFT. Evans’ 1979 article [1] constitutes a central reference to DFT. The proof of the variational principle that underlies the theory, including the existence and the uniqueness of the free energy functional, can be viewed as the classical analogue of Mermin’s earlier 1965 work on quantum systems at finite temperatures [2]. This forms a generalization of the Hohenberg-Kohn theorem for ground state properties of quantum systems. Historic milestones that predate these developments, and that can be re-formulated in classical DFT language, are Onsager’s 1949 treatment of the isotropic- nematic liquid crystal phase transition in systems of long and thin hard rods [3], and van der Waals’ 1893 theory of the microscopic structure of the liquid-gas interface [4]. Both the Hohenberg-Kohn proof and the Mermin-Evans proof start from a variational principle for the respective many-body function. In the quantum case, this is the Rayleigh-Ritz inequality for the many- body groundstate wave function. In the classical case, the theory rests on the Gibbs inequality for the equilibrium many-body phase space distribution. The corresponding functionals are the ground state energy in the quantum case and the thermodynamic grand potential in the classical case. Both functionals depend trivially on the position-dependent external one-body potential. Via an intricate sequence of arguments [1, 2], which has become textbook knowledge [5], the dependence on the external potential is played back to the more useful dependence on the in general position-dependent one-body density distribution. © M. Schmidt, M. Burgis, W.S.B. Dwandaru, G. Leithall, P. Hopkins, 2012 43603-1 M. Schmidt et al. In 1979 Levy showed that quantum DFT can be obtained in a more compact and straightforward way via a method that he called the constrained search [6] see also [7]. Here, the search for the minimum is performed in the space of all trial many-body wave functions. The constraint that is fixed during this search is that all trial wave functions considered need to generate the given one-body density distribu- tion. Levy’s derivation is both rigorous and elegant and constitutes a standard reference for electronic structure DFT. Among the impressive number of citations of his 1979 article, there are only very few pa- pers that draw connections to classical DFT, [8] being an example, although the method permits rather straightforward application to the classical case [9]. See also Percus’ general concept of “overcomplete” density functionals [10]. The constrained search method offers two significant benefits over the Mermin-Evans proof. One is simplicity, avoiding the reductio ad absurdum chain of arguments. The other is that it yields an explicit definition of the Helmholtz free energy density as a functional of the trial one-body density distribution. This formula, as reproduced in 2.2 below, is explicitly independent of the external potential. The gener- alization that facilitates this development is the concept of minimization in the space of all phase space distributions under the constraint of a given one-body density 2.4. The issue of representability of the one-body density has been addressed in [11]. It turns out that the simplicity of Levy’s method allows one to construct more general DFTs. An example is the variational framework developed in [12], which rests on the internal energy functional rather than the Helmholtz free-energy functional, which depends on the one-body density and on a local position-dependent entropy distribution. A dynamical version of this theory, based on linear irreversible thermodynamics and phenomenological reasoning, is proposed in [12]. Several common approximations for free energy functionals were transformed to internal en- ergy functionals. Having reliable approximations for the functional is a prerequisite for applying DFT to realistic three-dimensional model systems. The task of constructing usable functionals is different from the conceptual work outlined so far. In particular, a calculation of the constrained search expressions for the free-energy or internal-energy functional would amount to an exact solution of the many-body problem. Hence, the importance of these expressions is rather of conceptual nature. In 1989 Rosenfeld wrote a remarkable letter in which he proposed an approximate free energy func- tional for additive hard sphere mixtures [13]. His theory unified several earlier liquid state theories, such as the Percus-Yevick integral equation theory for the bulk structure [5], the scaled-particle theory for thermodynamics, and Rosenfeld’s own concepts, such as the scaled-field particle theory of [13], and encapsulated these into what he called fundamental measure theory FMT. Kierlik and Rosinberg in 1990 [14] re-wrote the same functional [15] in an alternative way, using only four not six, as Rosenfeld weight functions to build weighted densities via convolution with the bare one-body density of each hard sphere species. The two strands of FMT were pursued both with significant rigour and effort, see the re- cent reviews [16–18]. Rosenfeld’s more geometric approach was extended to further weight functions by Tarazona [19] in his treatment of freezing. A critical discussion of the properties of the relevant convolu- tion kernels can be found in [20]. Very recently, Korden [21, 22] demonstrated the relationship of the FMT with the exact virial expansion. It is vexing that Rosenfeld himself, who was certainly very versed in the diagrammatic techniques of liquid state theory see e.g. [23], apparently neither analysed nor formulated his very own FMT within such a framework. The non-local structure of FMT, its coupling of space integrals via convolution, and the plentiful appearances of the one-body density distributions seem to constitute an ideal playground for formulation in diagrammatic language. A comprehensive understanding of the diagrammatic nature of FMT could not help just to ascertain and clarify the nature of the approximations that are involved [21], but also, and more importantly from a pragmatic point of view, could enable one to construct new func- tionals for further model systems. Taking the FMT weight functions as bonds in a diagrammatic formu- lation, one immediately faces the combinatorial problems associated with their number, which in the Rosenfeld-Tarazona formulation are at least seven per hard sphere species four scalar, two vector, one tensor, which makes the book-keeping task a seemingly daunting one. It was shown [24, 25] that one can formulate the Kierlik-Rosinberg form of FMT in a diagrammatic way. The concept was applied to one-dimensional hard rods, where it gives Percus’ exact result [26, 27], as well as to five-dimensional hard hypersphere mixtures, where it gives a functional that outperforms Percus-Yevick theory, and improves on previous FMT attempts [28]. Two crucial properties of the dia- grammatic formulation can be identified. One is the relative simplicity of the diagrams that describe 43603-2 Recent developments in classical density functional theory the coupling of the various space integrals in the density functional. The topology of the diagrams is of star-like or tree-like shape, hence providing a significant reduction as compared to the complexity of the exact virial series. The number of arms is equal to the power in density and the bonds are weight func- tions rather than Mayer functions, like they are in the exact virial expansion. The book-keeping problem of having to deal with a large number of different weight functions is addressed and rendered almost trivial by exploiting the tensorial structure that underlies the Kierlik-Rosinberg form of FMT [29], where the geometric index of the weight functions is a proper tensorial index, and corresponding isometric and metamorphic transformations can be formulated [29]. Hence, the fully scalar Kierlik-Rosinberg formu- lation turned out to be indeed simpler to handle, and easier to generalize for full control of the degree of non-locality in the functional. These developments facilitated the generalization of FMT for binary non-additive hard spheres [30] to ternary mixtures [31]. In the present contribution, we describe the basic ideas underlying the above developments, without the full detail that is given in the respective original papers, but with further illustrative examples in order to provide an introduction to the subject. The paper is organized as follows. In section 2, Levy’s method is sketched, and both the free-energy and the internal-energy functionals are defined. A basic introduction to the diagrammatic formulation of FMT is given in section 3, including a brief overview of applications to non-additive hard sphere fluids in bulk and at interfaces. Section 4 gives concluding remarks.

2. Levy’s constrained search in classical DFT